Natural numbers:
Set of all numbers which is beginning with 1, 2,... is called as Natural numbers
Symbol for natural number = ℕ
Whole numbers:
Set of all numbers which is beginning with 0, 1, 2, 3...is called as whole numbers.
Symbol for natural number = W
Integers:
Integers are set of all whole numbers and their opposites. We are using number line to denote integers....-3, -2, -1, 0, 1, 2, 3....
Symbol for integers = ℤ
Rational numbers
In our number system we are representing the rational number in the form of fraction like a/b.
1.5 = 3/2
Symbol for rational number = ℚ
Irrational numbers
An irrational number is any real number which cannot be expressed as a simple fraction or rational number. Π, √2 are some examples or irrational numbers.
Symbol for rational number = R - ℚ
Real numbers
All numbers including positive integers, negative integers, rational numbers, irrational numbers, are called real numbers.
It is usually denoted as ℝ
Question 1 :
Classify each element of {√7, −1/4 , 0, 3.14, 4, 22/7} as a member of N, Q, R − Q or Z.
Solution :
√7 is irrational number
√7 ∈ R - ℚ
−1/4 is a rational number
−1/4 ∈ R - ℚ
0 is a integer
0 ∈ Z
3.14 is a rational number
3.14 ∈ Q
4 is a integer
4 ∈ Z
22/7 is a rational number.
22/7 ∈ Q
Question 2 :
Prove that √3 is an irrational number.
(Hint: Follow the method that we have used to prove √2 ∉ Q.)
Solution :
Let √3 be a rational number
√3 = p/q
p and q are co primes.
Co prime means the numbers which has common divisor other than 1.
p = √3 q
Taking squares on both sides, we get
p^{2} = (√3 q)^{2}
p^{2} = 3 q^{2}
q^{2 }= p^{2}/3
p is the divisor of 3.
p/3 = a
p = 3a
taking squares on both sides, we get
p^{2} = 9a^{2}
Here p^{2} = 3q^{2}
(3q^{2}) = 9a^{2}
q^{2} = 9a^{2}/3
a^{2 }= q^{2}/3
q is the factor of 3.
This contradicts out assumption. Hence √3 is irrational number.
Question 3 :
Are there two distinct irrational numbers such that their difference is a rational number? Justify.
Solution :
Let the two irrational numbers be 5 + √3 and 5 - √3.
Difference = (5 + √3) - (5 - √3)
= 5 + √3 - 5 + √3
= 2√3
Hence two distinct irrational numbers such that their difference is a rational number
Question 4 :
Find two irrational numbers such that their sum is a rational number. Can you find two irrational numbers whose product is a rational number.
Solution :
Let the two irrational numbers be 5 + √3 and 5 - √3.
Sum = (5 + √3) + (5 - √3)
= 5 + √3 + 5 - √3
= 10
Product = (5 + √3) (5 - √3)
= 5^{2} - √3^{2}
= 25 - 3
= 22
Question 5 :
Find a positive number smaller than 1/2^{1000} . Justify.
Solution :
We see
1/2^{1} > 1/2^{2} > 1/2^{3}
Hence the positive number less than 1/2^{1000 }is 1/2^{1001}
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